Characterization of a petroleum reservoir is a delicate problem faced by geophysicists, geologists and reservoir engineers. When one tries to clarify or to validate the description of a geologic model, interpretation of well tests, of interference tests and of production results plays a very important part.
Some of these interpretation techniques are based on inversion procedures where one tries to adjust numerical simulations to field measurements, from a gridded geologic model by varying the parameters of this model.
The use of numerical simulation has thus become more and more important, and as the geologic description has become increasingly finer, the size of the simulated models has increased, and calculating times, which increase correlatively, have remained very high despite computer performance improvements.
Similarly, during an inversion procedure using simulations on gridded models, it is essential to obtain very accurate solutions which introduce no numerical biases when they arc compared with field measurements.
In order to model flows in a petroleum reservoir, it is conventional to use the generalized Darcy equations, the law of mass conservation, and to associate therewith well conditions, boundary conditions and initial conditions.
A system of non-linear partial differential equations is thus obtained, whose unknowns are the pressures P and the saturations S of each phase, and possibly the concentrations of each constituent in the phases. ##EQU1## where .phi. is the porosity, C.sub.ip the concentration of constituent i, S.sub.p the saturation of phase p, .rho..sub.p the density and .delta..sub.qi the flow rate of constituent injected into or produced from the wells.
The velocity of the fluid and the pressure are related by Darcy's law: ##EQU2## where K is the absolute permeability, kr.sub.p the relative permeability of phase p; and .mu..sub.p the relative viscosity of phase p.
The saturation and the constituent balances, as well as the capillarity relations, are to be added to these equations.
In case of a barely compressible single-phase fluid, the system of equations reduces to: ##EQU3## with zero flow conditions or pressure conditions imposed at the edges and initial conditions. C represents the total compressibility (pores and fluid), supposedly constant in the model, and V .phi. the pore volume.
After discretization of the underground environment studied by means of a grid pattern, using a finite-volume solution method, the system of equations to be solved for each grid of the gridded model is: ##EQU4##
If the system is written in matrical form by denoting the transmissivity matrix by A, the diagonal matrix of CV .phi. by D and the second member by Q, we obtain: ##EQU5## where P is a vector of dimension n the number of unknowns which are the bottomhole pressures and in each grid cell of the gridded model, and P.sub.0 represents the initial state of the reservoir. Matrix A is a positive symmetrical square matrix nxn.
Diagonal matrix D being positive and independent of time, the matrical system can be written as follows: ##EQU6##
Solution for all the grid cells of the system of equations turns out to be very heavy and costly in calculating time if the number of grid cells of the model is very high, which is generally the case.